A350014 Numbers whose square has a number of divisors coprime to 6.

1, 4, 8, 9, 25, 27, 32, 36, 49, 64, 72, 100, 108, 121, 125, 169, 196, 200, 216, 225, 243, 256, 288, 289, 343, 361, 392, 441, 484, 500, 512, 529, 576, 675, 676, 729, 800, 841, 864, 900, 961, 968, 972, 1000, 1089, 1125, 1156, 1225, 1323, 1331, 1352, 1369, 1372, 1444

Offset

1,2

Comments

a(n) = m in A001694 such that d(m^2) is not divisible by 3, where d(n) = A000005(n).

Supersequence of A051676 (composite numbers whose square has a prime number of divisors).

Subsequence of A001694 (powerful numbers).

Numbers whose prime factorization has only exponents that are congruent to {0, 2} mod 3 (A007494). - Amiram Eldar, Mar 31 2022

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Formula

a(n) = {m : gcd(d(m^2), 6) = 1}.

Sum_{n>=1} 1/a(n) = 15*zeta(3)/Pi^2 (= 10 * A240976). - Amiram Eldar, Mar 31 2022

Maple

A350014 := proc(n)
    option remember ;
    local a;
    if n =1 then
        1;
    else
        for a from procname(n-1)+1 do
            if igcd(numtheory[tau](a^2), 6) = 1 then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A350014(n), n=1..20) ; # R. J. Mathar, Apr 06 2022

Mathematica

Select[Range[1500], CoprimeQ[DivisorSigma[0, #^2], 6] &] (* or *)
With[{nn = 1500}, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], Mod[DivisorSigma[0, #^2], 3] != 0 &]]

Prog

(PARI) isok(m) = gcd(numdiv(m^2), 6) == 1; \\ Michel Marcus, Mar 04 2022

Crossrefs

Cf. A000005, A000290, A007494, A001694, A001651, A051676, A240976.

Sequence in context: A063907 A261987 A158340 * A259183 A056166 A093771

Adjacent sequences: A350011 A350012 A350013 * A350015 A350016 A350017

Keyword

nonn,easy

Author

Michael De Vlieger, Jan 17 2022

Status

approved